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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 | import Mathlib
import OAI.Probability.BinarySweep.SparseBounds.ManySparse
import OAI.Probability.BinarySweep.GridBounds.BalancedGrid
namespace OAI
noncomputable section
section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps
open Young Irrep Representation Signed
def SpechtMomentBound (q : ℕ) {b : ℕ} (bits : Fin b → ℕ) (z : ℝ) : Prop :=
∀h : ℕ, ∀H : PathFamily bits h, ∀μ : YoungDiagram, ∀e : Cell μ ≃ FreeSlot H 0,
evenMoment q (groupAverage ((hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom)
(fun g => (conditionalGroupLaw H z g:ℂ))) ≤
Real.exp (-cExponent (gridSize bits)*diagramF μ+
eExponent (gridSize bits)*h*Real.log (gridSize bits)-pathCost H)
lemma spechtMomentBound_enumerated {q b : ℕ} {bits : Fin b → ℕ} {z : ℝ}
(hB : SpechtMomentBound q bits z) : EnumeratedMomentBound q bits z := by
intro h H α
let e : Cell (diagram α) ≃ FreeSlot H 0 :=
(cellsEquivFin α).trans (Fintype.equivFin _).symm
have hb := hB h H (diagram α) e
have he : groupAverage ((hilbertSpecht (diagram α)).comp e.symm.permCongrHom.toMonoidHom)
(fun g => (conditionalGroupLaw H z g:ℂ)) =
groupAverage (partitionHilbertRep α)
(fun a => (conditionalGroupLaw H z ((Fintype.equivFin _).symm.permCongr a):ℂ)) := by
rw [groupAverage_comp_slots]
symm
rw [partitionHilbertRep,groupAverage_comp_slots]
rfl
rw [he] at hb
simpa only [typeF,hilbertSpecht_finrank,diagramF] using hb
lemma hilbert_specht_complete {X V : Type*} [Fintype X] [DecidableEq X]
[NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
(ρ : Representation ℂ (Equiv.Perm X) V) [ρ.IsIrreducible] :
∃(μ : YoungDiagram) (e : Cell μ ≃ X), Nonempty (ρ.Equiv
((hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom)) := by
obtain ⟨μ,e,⟨f⟩⟩ := specht_complete ρ
let σ : Representation ℂ (Equiv.Perm X) (SpechtSpace μ) := (spechtRep μ).comp e.symm.permCongrHom.toMonoidHom
let τ : Representation ℂ (Equiv.Perm X) (SpechtHilbert μ) := (hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom
have a : σ.Equiv τ := .mk (spechtHilbertEquiv μ) (by
intro g
apply LinearMap.ext
intro v
change spechtHilbertEquiv μ (spechtRep μ _ v)=
spechtHilbertEquiv μ (spechtRep μ _ ((spechtHilbertEquiv μ).symm (spechtHilbertEquiv μ v)))
rw [LinearEquiv.symm_apply_apply])
exact ⟨μ,e,⟨f.trans a⟩⟩
lemma spechtMomentBound_main {q b : ℕ} {bits : Fin b → ℕ} {z : ℝ}
(hB : SpechtMomentBound q bits z) (h D : ℕ) (H : PathFamily bits h)
(ρ : Representation ℂ (Equiv.Perm (FreeSlot H 0)) (RepSpace D))
[ρ.IsIrreducible] (hρ : IsUnitaryRep ρ) :
logMoment (traceMoment q (conditionalOperator H z ρ)) ≤
((-cExponent (gridSize bits)*Real.log D+
eExponent (gridSize bits)*h*Real.log (gridSize bits)-pathCost H:ℝ):EReal) := by
obtain ⟨μ,e,⟨f⟩⟩ := hilbert_specht_complete ρ
have he := equivalent_evenMoment _ _ f hρ (fun g v => hilbertSpecht_unitary μ _ v)
(fun g => (conditionalGroupLaw H z g:ℂ)) q
have hD := f.toLinearEquiv.finrank_eq
rw [finrank_euclideanSpace_fin,hilbertSpecht_finrank] at hD
apply (logMoment_le_iff_le_exp _ _).mpr
rw [conditionalMoment_evenMoment,he]
simpa only [hD,diagramF] using hB h H μ e
end BinaryCoordinateSweeps
end
open scoped BigOperators Classical
open Filter
namespace BinaryCoordinateSweeps
open Young Irrep Representation GridSplit
theorem conditional_moment_main : ConditionalMomentTarget := by
obtain ⟨S,hS⟩ := sparse_closing_radius
have hq : 2 ≤ mainMomentPower := by norm_num [mainMomentPower]
obtain ⟨T,hT⟩ := eventually_atTop.mp (dense_induction_step_eventually mainMomentPower hq)
let A : ℝ := max ((10:ℝ)^26) ((max S T:ℕ):ℝ)
have hlog2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
obtain ⟨r,hr⟩ := exists_nat_gt (max (2:ℝ) (A/Real.log 2))
have hr2 : 2 ≤ r := by
have : (2:ℝ) < r := (le_max_left _ _).trans_lt hr
exact_mod_cast this.le
have hrA : A ≤ (r:ℝ)*Real.log 2 := by
have : A/Real.log 2 < r := (le_max_right _ _).trans_lt hr
exact (div_lt_iff₀ hlog2).mp this |>.le
have hrL : (10:ℝ)^26 ≤ (r:ℝ)*Real.log 2 := (le_max_left _ _).trans hrA
have hrM : ((max S T:ℕ):ℝ) ≤ (r:ℝ)*Real.log 2 := (le_max_right _ _).trans hrA
have hr4 : (4000000000:ℝ) ≤ (r:ℝ)*Real.log 2 := by linarith
obtain ⟨δ,hδ,hδline,hsp⟩ := hS r hr2 hr4
let zStar := min δ (oneAxisRadius r)
have hzStar : 0 < zStar := lt_min hδ (oneAxisRadius_pos r)
have hlarge {b : ℕ} (hb : 1 ≤ b) (bits : Fin b → ℕ)
(hd : ∀j,r ≤ bits j ∧ bits j ≤ 2*r) :
max S T ≤ gridSize bits ∧ (10:ℝ)^26 ≤ Real.log (gridSize bits) := by
have hl := grid_log_lower (fun j => (hd j).1)
have hbR : (1:ℝ) ≤ b := by exact_mod_cast hb
have hr0 : 0 ≤ (r:ℝ)*Real.log 2 := by positivity
have hbase : (r:ℝ)*Real.log 2 ≤ Real.log (gridSize bits) := by
have := mul_le_mul_of_nonneg_right hbR hr0
nlinarith
refine ⟨?_,hrL.trans hbase⟩
have hpos : (0:ℝ) < gridSize bits := by exact_mod_cast gridSize_pos bits
have hexp := Real.log_le_sub_one_of_pos hpos
have : ((max S T:ℕ):ℝ) ≤ gridSize bits := by linarith [hrM.trans hbase]
exact_mod_cast this
have hproof : ∀b : ℕ, 1 ≤ b → ∀bits : Fin b → ℕ,
(∀j,r ≤ bits j ∧ bits j ≤ 2*r) → ∀z : ℝ, 0 ≤ z → z ≤ zStar →
SpechtMomentBound mainMomentPower bits z := by
intro b
induction b using Nat.strong_induction_on with
| h b ih =>
intro hb bits hd z hz hzr h H μ e
by_cases hb1 : b=1
· subst b
have := irreducible_comp_equiv e.symm.permCongrHom (hilbertSpecht μ)
have hbase := hilbert_conditional_one_axis (by linarith : (400000000:ℝ) ≤ (r:ℝ)*Real.log 2)
(by omega : 0 < mainMomentPower) bits hd H hz (hzr.trans (min_le_right _ _))
((hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom)
(fun g v => hilbertSpecht_unitary μ _ v)
simpa only [hilbertSpecht_finrank,diagramF] using hbase
· have hb2 : 2 ≤ b := by omega
obtain ⟨m,n,rfl,hm,hn,hmb,hnb,hmn,hnm⟩ := balanced_counts b hb2
obtain ⟨hsize,hL⟩ := hlarge hb bits hd
by_cases hF : (gridSize bits:ℝ)^(399/400:ℝ) ≤ diagramF μ
· obtain ⟨hu,hv,hub,hvb,haxes⟩ := balanced_grid_bounds bits hm hn hmn hnm hd (by linarith)
apply hT (gridSize bits) ((le_max_right S T).trans hsize) m n h r bits H rfl
hu hv hub hvb (fun j => (hd j).2) haxes μ e hF z hz
((hzr.trans (min_le_left _ _)).trans hδline)
· apply spechtMomentBound_enumerated
exact ih m hmb hm (leftBits bits) (fun j => hd (j.castAdd n)) z hz hzr
· apply spechtMomentBound_enumerated
exact ih n hnb hn (rightBits bits) (fun j => hd (j.natAdd m)) z hz hzr
· exact hsp (gridSize bits) ((le_max_left S T).trans hsize) (m+n) h bits rfl hb hd hL H μ e
(lt_of_not_ge hF) z hz (hzr.trans (min_le_left _ _))
refine ⟨r,by omega,mainMomentPower,by omega,zStar,hzStar,?_⟩
intro b hb bits hd h H z hz hzr D ρ hi hρ
have := hi
exact spechtMomentBound_main (hproof b hb bits hd z hz hzr) h D H ρ hρ
end BinaryCoordinateSweeps
end
end OAI
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